Computer Architecture CST 250 Number Systems in Brief Prepared by:Omar Hirzallah

Contents The Number Systems Conversions ASCII Coding BCD Address Range Binary Numbers Binary Arithmetic (Add. & Sub.) S & M, 1’s & 2’s Complement Methods

THE NUMBER SYSTEM (1) Binary Number System (0,1) Base 2 (2) Octal Number System (0,1,2,3,4,5,6,7) Base 8 (3) Decimal Number System (Denary) (0,1,2,3,4,5,6,7,8,9) Base 10 (4) Hexadecimal Number System (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) Base 16

The ASCII Code ASCII stands for “American Standard Codes for Information Interchange”. H= 72 0 1 0 0 1 0 0 0 e= 101 0 1 1 0 0 1 0 1 l = 108 0 1 1 0 1 1 0 0 l = 108 0 1 1 0 1 1 0 0 o= 111 0 1 1 0 1 1 1 1 It’s a 7 + 1 bit method. 7 bits for code values and 1 bit for Parity check.

Binary Coded Decimal Decimal Symbol BCD Digit 0000 1 0001 2 0010 3 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001

8 7 1000 0111 The Binary Coded Decimals (BCD) ASCII Codes use 1 byte for 1 character to store Whereas BCD can be used to save memory space by putting two characters in one byte. Example: 87 can be written as 8 7 1000 0111

No. of Different Codes = 2n THE ADDRESS RANGE: The Formula to calculate the no. of different combinations /addresses range according to the no. of bits: No. of Different Codes = 2n (Where n is no. of bits.) For Example: If there are two bits then: No. of Different Codes = 22 = 2 x 2 = 4 00 01 10 11 For Example: If there are three bits then: No. of Different Codes = 23 = 2 x 2 x 2 = 8 000 001 010 011 100 101 110 111

The Binary Numbers The Example of Unsigned Binary Numbers: Signed Binary Numbers (+ or -) The Example of Unsigned Binary Numbers: 11001 Decimal Equivalent 25 M.S.B. L.S.B. M.S.B. Most Significance Bit L.S.B. Least Significance Bit

Signed (- or +) binary numbers There are two very famous notations in dealing with Sign & Magnitude Method (7 + 1 bit method): 0 0 0 0 0 0 1 1 + 3 1 0 0 0 0 0 1 1 - 3 Sign Bit Magnitude Complement Method (2’s complement): 2’s Complement = 1’s complement +1 1’s complement : Convert all 1s to 0s and a 0s to 1s

1’s Complement : Convert all 0’s into 1’s and all 1’s into 0’s 1’s Complement : Convert all 0’s into 1’s and all 1’s into 0’s. For Example: 0 0 1 0 0 1 0 1 1’s Complement: 1 1 0 1 1 0 1 0 2’s Complement : Convert all 0’s into 1’s and all 1’s into 0’s and then add 1. For Example: 0 0 1 0 0 1 0 1 (37) 1’s Complement: 1 1 0 1 1 0 1 0 __________1 2’s Complement: 1 1 0 1 1 0 1 1 (-37) +

BINARY ARITHMETIC: BINARY ADDITION: There are four Basic Rules for Binary Addition: 0 0 1 1 0+ 1+ 0+ 1+ 0 1 1 10 FOR EXAMPLE: 1 0 0 1 1 1 0 1 0 1 1 0 + 1 1 1 1 + 1 1 1 1 1 1 1 0 0 1 1 1 1 1

BINARY ARITHMETIC: 0 1 1 0 - 1 1 BINARY SUBTRACTION: 1)There are four Basic Rules for Binary Subtraction: 0 1 1 0 0- 0- 1- 1- 0 1 0 Borrow FOR EXAMPLE: 1 0 0 1 0 1 1 0 - If any, otherwise impossible to solve 1 10 10 1 1

BINARY ARITHMETIC: =0101 + (-0011) =0101 + (1100+1) =0101 + 1101 BINARY SUBTRACTION: 2-Use the 2’s complement method: FOR EXAMPLE: (a 4-bit number) A – B can be expressed as A + (-B) (-B) is the 2’s complement of B 0 1 0 1 – 0 0 1 1 =0101 + (-0011) =0101 + (1100+1) =0101 + 1101 =1 0010 (5) (3) (2)

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